PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 137, Number 10, October 2009, Pages 3497–3510
S 0002-9939(09)09945-6
Article electronically published on May 28, 2009
ON NOTIONS OF HARMONICITY
ZHEN-QING CHEN
(Communicated by Edward C. Waymire)
Abstract. In this paper we address the equivalence of the analytic and probabilistic notions of harmonicity in the context of general symmetric Hunt processes on locally compact separable metric spaces. Extensions to general symmetric right processes on Lusin spaces, including infinite dimensional spaces,
are mentioned at the end of this paper.
1. Introduction
It is known that a function 𝑢 being harmonic in a domain 𝐷 ⊂ ℝ𝑛 can be defined
1,2
(𝐷) :=
or characterized by Δ𝑢 = 0 in}𝐷 in the distributional sense; that is, 𝑢 ∈ 𝑊loc
{
2
2
𝑣 ∈ 𝐿loc (𝐷) ∣ ∇𝑣 ∈ 𝐿loc (𝐷) so that
∫
∇𝑢(𝑥) ⋅ ∇𝑣(𝑥)𝑑𝑥 = 0
for every 𝑣 ∈ 𝐶𝑐∞ (𝐷).
ℝ𝑛
This is equivalent to the following averaging property by running a Brownian motion
𝑋: for every relatively compact subset 𝑈 of 𝐷:
𝑢(𝑋𝜏𝑈 ) ∈ 𝐿1 (P𝑥 )
and
𝑢(𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )]
for every 𝑥 ∈ 𝑈.
/ 𝑈 }. Recently there has been interest (e.g. [2]) from
Here 𝜏𝑈 := inf {𝑡 ≥ 0 : 𝑋𝑡 ∈
several areas of mathematics in determining whether the above two notions of harmonicity remain equivalent in a more general context, such as for diffusions on
fractals (see [1]) and for discontinuous processes including symmetric Lévy processes. For instance, due to their importance in theory and in applications, there
has been intense interest recently in studying discontinuous processes and non-local
(or integro-differential) operators by both analytical and probabilistic approaches.
See, e.g., [5, 6] and the references therein. So it is important to identify the connection between the analytic and probabilistic notions of harmonic functions.
In this paper, we address the question of the equivalence of the analytic and
probabilistic notions of harmonicity in the context of symmetric Hunt processes on
local compact separable metric spaces. Let 𝑋 be an 𝑚-symmetric Hunt process on
a locally compact separable metric space 𝐸 whose associated Dirichlet form (ℰ, ℱ)
is regular on 𝐿2 (𝐸; 𝑚). Let 𝐷 be an open subset of 𝐸 and 𝜏𝐷 be the first exit
time from 𝐷 by 𝑋. Motivated by the example at the beginning of this section,
Received by the editors October 25, 2008, and, in revised form, February 16, 2009.
2000 Mathematics Subject Classification. Primary 60J45, 31C05; Secondary 31C25, 60J25.
Key words and phrases. Harmonic function, uniformly integrable martingale, symmetric Hunt
process, Dirichlet form, Lévy system.
The research of this author is supported in part by NSF Grant DMS-0600206.
c
⃝2009
Zhen-Qing Chen
3497
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3498
ZHEN-QING CHEN
loosely speaking (see the next section for precise statements), there are two ways to
define a function 𝑢 being harmonic in 𝐷 with respect to 𝑋: (a) (probabilistically)
𝑡 → 𝑢(𝑋𝑡∧𝜏𝐷 ) is a P𝑥 -uniformly integrable martingale for quasi-every 𝑥 ∈ 𝐷; (b)
(analytically) ℰ(𝑢, 𝑔) = 0 for 𝑔 ∈ ℱ ∩ 𝐶𝑐 (𝐷). We will show in Theorem 2.11 below
that these two definitions are equivalent. Note that even in the Brownian motion
case a function 𝑢 that is harmonic in 𝐷 is typically not in the domain ℱ of the
𝐷
the family of functions 𝑢 on 𝐸 such that for every
Dirichlet form. Denote by ℱloc
relatively compact open subset 𝐷1 of 𝐷, there is a function 𝑓 ∈ ℱ so that 𝑢 = 𝑓 𝑚a.e. on 𝐷1 . To show these two definitions are equivalent, the crux of the difficulty
is to
𝐷
(i) appropriately extend the definition of ℰ(𝑢, 𝑣) to functions 𝑢 in ℱloc
that
satisfy some minimal integrability condition when 𝑋 is discontinuous so
that ℰ(𝑢, 𝑣) is well defined for every 𝑣 ∈ ℱ ∩ 𝐶𝑐 (𝐷);
𝐷
(ii) show that if 𝑢 is harmonic in 𝐷 in the probabilistic sense, then 𝑢 ∈ ℱloc
and ℰ(𝑢, 𝑣) = 0 for every 𝑣 ∈ ℱ ∩ 𝐶𝑐 (𝐷).
If one assumes a priori that 𝑢 ∈ ℱ, then the equivalence of (a) and (b) is easy to
establish. See Remarks 2.8(i) and 2.10 below.
In next section, we give precise definitions and statements of the main results
and their proofs. Three examples are given to illustrate the main results of this
paper. Extensions to general symmetric right processes on Lusin spaces including
infinite dimensional spaces are mentioned at the end of this paper. We use “:=” as
a means of definition. For two real numbers 𝑎 and 𝑏, 𝑎 ∧ 𝑏 := min{𝑎, 𝑏}.
2. Main results
Let 𝑋 = (Ω, ℱ∞ , ℱ𝑡 , 𝑋𝑡 , 𝜁, P𝑥 , 𝑥 ∈ 𝐸) be an 𝑚-symmetric Hunt process on a
locally compact separable metric space 𝐸, where 𝑚 is a positive Radon measure on
𝐸 with full topological support. A cemetery state ∂ is added to 𝐸 to form 𝐸∂ :=
𝐸 ∪ {∂} as its one-point compactification, and Ω is the totality of right-continuous,
left-limited sample paths from [0, ∞[ to 𝐸∂ that hold the value ∂ once attaining
it. For any 𝜔 ∈ Ω, we set 𝑋𝑡 (𝜔) := 𝜔(𝑡). Let 𝜁(𝜔) := inf{𝑡 ≥ 0 ∣ 𝑋𝑡 (𝜔) = ∂}
be the lifetime of 𝑋. As usual, ℱ∞ and ℱ𝑡 are the minimal augmented 𝜎-algebras
0
:= 𝜎{𝑋𝑠 ∣ 0 ≤ 𝑠 < ∞} and ℱ𝑡0 := 𝜎{𝑋𝑠 ∣ 0 ≤ 𝑠 ≤ 𝑡} under
obtained from ℱ∞
/ 𝐵} (the exit time
{P𝑥 : 𝑥 ∈ 𝐸}. For a Borel subset 𝐵 of 𝐸, 𝜏𝐵 := inf{𝑡 > 0 ∣ 𝑋𝑡 ∈
of 𝐵) and 𝜎𝐵 := inf{𝑡 ≥ 0 ∣ 𝑋𝑡 ∈ 𝐵} (the entrance time of 𝐵) are (ℱ𝑡 )-stopping
times.
The transition semigroup {𝑃𝑡 : 𝑡 ≥ 0} of 𝑋 is defined by
𝑃𝑡 𝑓 (𝑥) := E𝑥 [𝑓 (𝑋𝑡 )] = E𝑥 [𝑓 (𝑋𝑡 ) : 𝑡 < 𝜁],
𝑡 ≥ 0.
Each 𝑃𝑡 may be viewed as an operator on 𝐿2 (𝐸, 𝑚), and taken as a whole, these
operators form a strongly continuous semigroup of self-adjoint contractions. The
Dirichlet form associated with 𝑋 is the bilinear form
ℰ(𝑢, 𝑣) := lim 𝑡−1 (𝑢 − 𝑃𝑡 𝑢, 𝑣)𝑚
(2.1)
𝑡↓0
defined on the space
(2.2)
ℱ :=
{
}
−1
𝑢 ∈ 𝐿 (𝐸; 𝑚) sup 𝑡 (𝑢 − 𝑃𝑡 𝑢, 𝑢)𝑚 < ∞ .
2
𝑡>0
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ON NOTIONS OF HARMONICITY
3499
∫
Here we use the notation (𝑓, 𝑔)𝑚 := 𝐸 𝑓 (𝑥)𝑔(𝑥) 𝑚(𝑑𝑥). We assume that (ℰ, ℱ) is a
regular Dirichlet form on 𝐿2 (𝐸; 𝑚); that is, 𝐶𝑐 (𝐸)∩ℱ is dense both in (𝐶𝑐 (𝐸), ∥⋅∥∞ )
and in (ℱ, ℰ1 ). Here 𝐶𝑐 (𝐸) is the space of continuous functions with compact
support in 𝐸 and ℰ1 (𝑢, 𝑢) := ℰ(𝑢, 𝑢) + (𝑢, 𝑢)𝑚 . However to ensure a wide scope of
applicability, we do not assume that the process 𝑋 (or equivalently, its associated
Dirichlet form (ℰ, ℱ)) is 𝑚-irreducible.
We refer readers to [4] and [9] for the following known facts. The extended
Dirichlet space ℱ𝑒 is the space of all functions 𝑓 on 𝐸 so that there is an ℰ-Cauchy
sequence {𝑓𝑛 , 𝑛 ≥ 1} ⊂ ℱ so that 𝑓𝑛 converges to 𝑓 𝑚-a.e. on 𝐸. For such an
𝑓 ∈ ℱ𝑒 , ℰ(𝑓, 𝑓 ) := lim𝑛→∞ ℰ(𝑓𝑛 , 𝑓𝑛 ). Every 𝑓 ∈ ℱ𝑒 admits a quasi-continuous
version (cf. [9, Theorem 2.1.7]). Throughout this paper, we always assume that
every function in ℱ𝑒 is represented by its quasi-continuous version, which is unique
up to a set of zero capacity (that is, quasi-everywhere, or q.e. for abbreviation).
We adopt the convention that any function 𝑓 defined on 𝐸 is extended to 𝐸∂
by taking 𝑓 (∂) = 0 and that 𝑋∞ (𝜔) := ∂ for every 𝜔 ∈ Ω. It is known that
ℱ𝑒 ∩ 𝐿2 (𝐸; 𝑚] = ℱ. The extended Dirichlet form (ℰ, ℱ𝑒 ) admits the following
Beurling-Deny decomposition (cf. [4, Theorem 4.3.3] or [9, Theorem 5.3.1]):
∫
∫
1
(𝑐)
2
(𝑢(𝑥) − 𝑢(𝑦)) 𝐽(𝑑𝑥, 𝑑𝑦) +
𝑢(𝑥)2 𝜅(𝑑𝑥),
ℰ(𝑢, 𝑢) = ℰ (𝑢, 𝑢) +
2 𝐸×𝐸
𝐸
where ℰ (𝑐) is the strongly local part of (ℰ, ℱ), 𝐽 the jumping measure and 𝜅 the
killing measure of (ℰ, ℱ) (or of 𝑋). For 𝑢, 𝑣 ∈ ℱ𝑒 , ℰ (𝑐) (𝑢, 𝑣) can also be expressed
by the mutual energy measure 12 𝜇𝑐⟨𝑢,𝑣⟩ (𝐸), which is the signed Revuz measure
associated with 12 ⟨𝑀 𝑢,𝑐 , 𝑀 𝑣,𝑐 ⟩. Here for 𝑢 ∈ ℱ𝑒 , 𝑀 𝑢,𝑐 denotes the continuous
martingale part of the square integrable martingale additive functional 𝑀 𝑢 of 𝑋 in
the Fukushima’s decomposition (cf. [9, Theorem 5.2.2]) of
𝑢(𝑋𝑡 ) − 𝑢(𝑋0 ) = 𝑀𝑡𝑢 + 𝑁𝑡𝑢 ,
𝑡 ≥ 0,
where 𝑁 𝑢 is a continuous additive functional of 𝑋 having zero energy. When 𝑢 = 𝑣,
it is customary to write 𝜇𝑐⟨𝑢,𝑢⟩ as 𝜇𝑐⟨𝑢⟩ . The measure 𝜇𝑐⟨𝑢,𝑣⟩ enjoys the strong local
property in the sense that if 𝑢 ∈ ℱ𝑒 is constant on a nearly Borel quasi-open set
𝐷, then 𝜇𝑐⟨𝑢,𝑣⟩ (𝐷) = 0 for every 𝑣 ∈ ℱ𝑒 (see [4, Proposition 4.3.1]). For 𝑢 ∈ ℱ, let
𝜇⟨𝑢⟩ be the Revuz measure of ⟨𝑀 𝑢 ⟩. Then it holds that
∫
1
1
𝑢(𝑥)2 𝜅(𝑑𝑥).
ℰ(𝑢, 𝑢) = 𝜇⟨𝑢⟩ (𝐸) +
2
2 𝐸
For an open subset 𝐷 of 𝐸, we use 𝑋 𝐷 to denote the subprocess of 𝑋 killed upon
leaving 𝐷. The Dirichlet form of 𝑋 𝐷 on 𝐿2 (𝐷; 𝑚) is (ℰ, ℱ 𝐷 ), where ℱ 𝐷 := {𝑢 ∈
ℱ ∣ 𝑢 = 0 q.e. on 𝐷𝑐 }. It is known (cf. [4, Theorem 3.3.9] or [9, Theorem 4.4.3])
that (ℰ, ℱ 𝐷 ) is a regular Dirichlet form on 𝐿2 (𝐷; 𝑚). Let ℱ𝑒𝐷 := {𝑢 ∈ ℱ𝑒 ∣ 𝑢 =
0 q.e. on 𝐷𝑐 }. Then ℱ𝑒𝐷 is the extended Dirichlet space of (ℰ, ℱ 𝐷 ) (see Theo𝐷
, if
rem 3.4.9 of [4]). A function 𝑓 is said to be locally in ℱ 𝐷 , denoted as 𝑓 ∈ ℱloc
𝐷
for every relatively compact subset 𝑈 of 𝐷, there is a function 𝑔 ∈ ℱ such that
𝐷
𝑓 = 𝑔 𝑚-a.e. on 𝑈 . Every 𝑓 ∈ ℱloc
admits an 𝑚-version that is quasi-continuous
𝐷
when
on 𝐷. Throughout this paper, we always assume that every function in ℱloc
restricted to 𝐷 is represented by its quasi-continuous version. By the strong local
𝐷
. We
property of 𝜇𝑐⟨𝑢,𝑣⟩ for 𝑢, 𝑣 ∈ ℱ, 𝜇𝑐⟨𝑢,𝑣⟩ is well defined on 𝐷 for every 𝑢, 𝑣 ∈ ℱloc
∞
use 𝐿loc (𝐷; 𝑚) to denote the 𝑚-equivalent class of locally bounded functions on 𝐷.
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3500
ZHEN-QING CHEN
Let (𝑁 (𝑥, 𝑑𝑦), 𝐻) be a Lévy system of 𝑋 (cf. [3] or [9]). Then
𝐽(𝑑𝑥, 𝑑𝑦) = 𝑁 (𝑥, 𝑑𝑦)𝜇𝐻 (𝑑𝑥)
and
𝜅(𝑑𝑥) := 𝑁 (𝑥, ∂)𝜇𝐻 (𝑑𝑥),
where 𝜇𝐻 is the Revuz measure of the positive continuous additive functional 𝐻 of
𝑋.
Definition 2.1. Let 𝐷 be an open subset of 𝐸. We say a function 𝑢 is harmonic
in 𝐷 (with respect to the process 𝑋) if for every relatively compact open subset 𝑈
of 𝐷, 𝑡 → 𝑢(𝑋𝑡∧𝜏𝑈 ) is a uniformly integrable P𝑥 -martingale for q.e. 𝑥 ∈ 𝑈 .
To derive an analytic characterization of harmonic functions in 𝐷 in terms of an
extension of quadratic form (ℰ, ℱ), we need some preparation. Let 𝑟𝑡 denote the
time-reversal operator defined on the path space Ω of 𝑋 as follows: For 𝜔 ∈ {𝑡 < 𝜁},
{
𝜔((𝑡 − 𝑠)−) if 0 ≤ 𝑠 < 𝑡,
𝑟𝑡 (𝜔)(𝑠) =
𝜔(0)
if 𝑠 ≥ 𝑡.
(It should be borne in mind that the restriction of the measure P𝑚 to ℱ𝑡 is invariant
under 𝑟𝑡 on Ω ∩ {𝜁 > 𝑡}.)
Lemma 2.2. If 𝑢 ∈ ℱ𝑒 has ℰ(𝑢, 𝑢) = 0, then
P𝑥 (𝑢(𝑋𝑡 ) = 𝑢(𝑋0 ) for every 𝑡 ≥ 0) = 1
for q.e. 𝑥 ∈ 𝐸.
In other words, for q.e. 𝑥 ∈ 𝐸, 𝐸𝑥 := {𝑦 ∈ 𝐸 : 𝑢(𝑦) = 𝑢(𝑥)} is an invariant set
with respect to the process 𝑋 in the sense that P𝑥 (𝑋[0, ∞) ⊂ 𝐸𝑥 ) = 1. This in
particular implies that if, in addition, P𝑥 (𝜁 < ∞) > 0 for q.e. 𝑥 ∈ 𝐸, then 𝑢 = 0
q.e. on 𝐸.
Proof. It is known (see, e.g., [4, Theorem 6.6.2]) that the following Lyons-Zheng
forward-backward martingale decomposition holds for 𝑢 ∈ ℱ𝑒 :
1
1
𝑢(𝑋𝑡 ) − 𝑢(𝑋0 ) = 𝑀𝑡𝑢 − 𝑀𝑡𝑢 ∘ 𝑟𝑡 P𝑚 -a.e. on {𝑡 < 𝜁}.
2
2
As 𝜇⟨𝑢⟩ (𝐸) ≤ 2ℰ(𝑢, 𝑢) = 0, we have 𝑀 𝑢 = 0, and so 𝑢(𝑋𝑡 ) = 𝑢(𝑋0 ) P𝑚 -a.s. on
{𝑡 < 𝜁} for every 𝑡 > 0. This implies via Fukushima’s(decomposition that 𝑁 𝑢 = 0 on
[0, 𝜁) and hence) on [0, ∞) P𝑚 -a.s. Consequently, P𝑥 𝑢(𝑋𝑡 )−𝑢(𝑋0 ) = 𝑀𝑡𝑢 +𝑁𝑡𝑢 = 0
for every 𝑡 ≥ 0 = 1 for q.e. 𝑥 ∈ 𝐸. This proves the lemma.
□
Since (ℰ, ℱ) is a regular Dirichlet form on 𝐿2 (𝐸; 𝑚), for any relatively compact
open sets 𝑈, 𝑉 with 𝑈 ⊂ 𝑉 , there is 𝜙 ∈ ℱ ∩ 𝐶𝑐 (𝐸) so that 𝜙 = 1 on 𝑈 and 𝜙 = 0
on 𝑉 𝑐 . Consequently,
∫
(2.3)
𝐽(𝑈, 𝑉 𝑐 ) =
(𝜙(𝑥) − 𝜙(𝑦))2 𝐽(𝑑𝑥, 𝑑𝑦) ≤ 2ℰ(𝜙, 𝜙) < ∞.
𝑈×𝑉 𝑐
For an open set 𝐷 ⊂ 𝐸, consider the following two conditions for the function 𝑢
on 𝐸. For any relatively compact open sets 𝑈, 𝑉 with 𝑈 ⊂ 𝑉 ⊂ 𝑉 ⊂ 𝐷,
∫
(2.4)
∣𝑢(𝑦)∣𝐽(𝑑𝑥, 𝑑𝑦) < ∞
𝑈×(𝐸∖𝑉 )
and
(2.5)
1𝑈 (𝑥)E𝑥
[(
)
]
(1 − 𝜙𝑉 )∣𝑢∣ (𝑋𝜏𝑈 ) ∈ ℱ𝑒𝑈 ,
where 𝜙𝑉 ∈ 𝐶𝑐 (𝐷) ∩ ℱ with 0 ≤ 𝜙𝑉 ≤ 1 and 𝜙𝑉 = 1 on 𝑉 . Note that both
conditions (2.4) and (2.5) are automatically satisfied when 𝑋 is a diffusion, since
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ON NOTIONS OF HARMONICITY
3501
in this case the jumping measure 𝐽 vanishes and 𝑋𝜏𝑈 ∈ ∂𝑈 on {𝜏𝑈 < 𝜁}. In view
of (2.3), every bounded function 𝑢 satisfies condition (2.4). In fact by the following
lemma, every bounded function 𝑢 also satisfies condition (2.5).
Lemma 2.3. Suppose that 𝑢 is a function on 𝐸 satisfying condition (2.4) and that
for any relatively compact open sets 𝑈, 𝑉 with 𝑈 ⊂ 𝑉 ⊂ 𝑉 ⊂ 𝐷,
[(
)
]
(2.6)
sup E𝑥 1𝑉 𝑐 ∣𝑢∣ (𝑋𝜏𝑈 ) < ∞.
𝑥∈𝑈
Then (2.5) holds for 𝑢.
In many concrete cases such as in Examples 2.12-2.14 below, one can show that
condition (2.4) implies condition (2.6). To prove the above lemma, we need the
following result. Observe that the process 𝑋 is not assumed to be transient.
Lemma 2.4. Suppose that 𝜈 is a smooth measure on 𝐸 whose corresponding
positive continuous additive
functional (PCAF) of 𝑋 is denoted as 𝐴𝜈 . Define
∫
𝜈
𝐺𝜈(𝑥) := E𝑥 [𝐴𝜁 ]. If 𝐸 𝐺𝜈(𝑥)𝜈(𝑑𝑥) < ∞, then 𝐺𝜈 ∈ ℱ𝑒 . Moreover,
∫
(2.7)
ℰ(𝐺𝜈, 𝑢) =
𝑢(𝑥)𝜈(𝑑𝑥)
for every 𝑢 ∈ ℱ𝑒 .
𝐸
Proof. First assume that 𝑚(𝐸) < ∞. It is easy to check directly that {𝑥 ∈ 𝐸 :
E𝑥 [𝐴𝜈𝜁 ] > 𝑗} is finely open for every integer 𝑗 ≥ 1. So 𝐾𝑗 := {𝐺𝜈 ≤ 𝑗} is finely
∪
closed. Since 𝐺𝜈 < ∞ 𝜈-a.e. on 𝐸, we have 𝜈(𝐸 ∖ ∞
𝑗=1 𝐾𝑗 ) = 0. Define 𝜈𝑗 := 1𝐾𝑗 𝜈.
Clearly for 𝑥 ∈ 𝐾𝑗 , 𝐺𝜈𝑗 (𝑥) ≤ 𝐺𝜈(𝑥) ≤ 𝑗, while for 𝑥 ∈ 𝐾𝑗𝑐 ,
[∫
]
𝜁
[
]
𝜈
1𝐾𝑗 (𝑋𝑠 )𝑑𝐴𝑠 = E𝑥 𝐺𝜈𝑗 (𝑋𝜎𝐾𝑗 ) ≤ 𝑗.
𝐺𝜈𝑗 (𝑥) = E𝑥
0
So 𝑓𝑗 := 𝐺𝜈𝑗 ≤ 𝑗 on 𝐸 and hence is in 𝐿2 (𝐸; 𝑚). Since by [4, Theorem 4.1.1] or
[9, Theorem 5.1.3]
(2.8)
∫
∫
[ 𝜈𝑗 ]
1
1
𝑓𝑗 (𝑥)𝜈𝑗 (𝑑𝑥) ≤
𝐺𝜈(𝑥)𝜈(𝑑𝑥) < ∞,
lim (𝑓𝑗 − 𝑃𝑡 𝑓𝑗 , 𝑓𝑗 )𝑚 = lim E𝑓𝑗 ⋅𝑚 𝐴𝑡 =
𝑡→0 𝑡
𝑡→0 𝑡
𝐸
𝐸
∫
we have 𝑓𝑗 ∈ ℱ with ℰ(𝑓
[ 𝑗 , 𝑓𝑗 ) ≤ ]𝐸 𝐺𝜈(𝑥)𝜈(𝑑𝑥). The same calculation shows that
1𝐾 ∖𝐾 ⋅𝜈
for 𝑖 > 𝑗, 𝑓𝑖 − 𝑓𝑗 = E𝑥 𝐴𝜁 𝑖 𝑗
and
∫
∫
(𝑓𝑖 − 𝑓𝑗 )(𝑥)𝜈(𝑑𝑥) ≤
𝐺𝜈(𝑥)𝜈(𝑑𝑥),
ℰ(𝑓𝑖 − 𝑓𝑗 , 𝑓𝑖 − 𝑓𝑗 ) =
𝐾𝑖 ∖𝐾𝑗
𝐾𝑙 ∖𝐾𝑗
which tends to zero as 𝑖, 𝑗 → ∞; that is, {𝑓𝑗 , 𝑗 ≥ 1} is an ℰ-Cauchy sequence in ℱ.
As lim𝑗→∞ 𝑓𝑗 = 𝑓 on 𝐸, we conclude that 𝑓 ∈ ℱ𝑒 . We deduce from (2.8) that
∫
(2.9)
ℰ(𝑓, 𝑓 ) = lim ℰ(𝑓𝑗 , 𝑓𝑗 ) =
𝐺𝜈(𝑥)𝜈(𝑑𝑥).
𝑗→∞
𝐸
ℱ𝑏+ ,
Moreover, for 𝑢 ∈
by [4, Theorem 4.1.1] (or [9, Theorem 5.1.3]) and the
dominated convergence theorem, we have
1
ℰ(𝐺𝜈, 𝑢) = lim ℰ(𝑓𝑗 , 𝑢) = lim lim (𝑓𝑗 − 𝑃𝑡 𝑓𝑗 , 𝑢)
𝑗→∞
𝑗→∞ 𝑡→0 𝑡
∫
∫
= lim
𝑢(𝑥)1𝐾𝑗 (𝑥)𝜈(𝑑𝑥) =
𝑢(𝑥)𝜈(𝑑𝑥).
𝑗→∞
𝐸
𝐸
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3502
ZHEN-QING CHEN
Since the linear span of ℱ𝑏+ is ℰ-dense in ℱ𝑒 , we have established (2.7).
For a general 𝜎-finite measure 𝑚, take a strictly positive 𝑚-integrable Borel
measurable function 𝑔 on 𝐸 and define 𝜇 = 𝑔 ⋅ 𝑚. Then 𝜇 is a finite measure
on 𝐸. Let 𝑌 be∫ the time-change of 𝑋 via measure 𝜇; that is, 𝑌𝑡 = 𝑋𝜏𝑡 , where
𝑠
𝜏𝑡 = inf{𝑠 > 0 : 0 𝑔(𝑋𝑠 )𝑑𝑠 > 𝑡}. The time-changed process 𝑌 is 𝜇-symmetric. Let
𝑌
𝑌
(ℰ , ℱ ) be the Dirichlet form of 𝑌 on 𝐿2 (𝐸; 𝜇). Then it is known that ℱ𝑒𝑌 = ℱ𝑒
and ℰ 𝑌 = ℰ on ℱ𝑒 (see (5.2.17) of [4]). The measure 𝜈 is also a smooth measure
with respect to the process 𝑌 . It is easy to verify that the PCAF 𝐴𝑌,𝜈 of 𝑌
corresponding to 𝜈 is related to the corresponding PACF 𝐴𝜈 of 𝑋 by
𝐴𝑌,𝜈
= 𝐴𝜈𝜏𝑡
𝑡
for 𝑡 ≥ 0.
In particular, we have 𝐺𝑌 𝜈(𝑥) = 𝐺𝜈 on 𝐸. Since we just proved that the lemma
holds for 𝑌 , we conclude that the lemma also holds for 𝑋.
□
Proof of Lemma 2.3. For relatively compact open sets 𝑈 , 𝑉 with 𝑈 ⊂ 𝑉 ⊂ 𝑉 ⊂
(𝐷) with
𝐷 and 𝜙[(
𝑉 ∈ ℱ ∩ 𝐶𝑐)
] 0 ≤ 𝜙𝑉 ≤ 1 and 𝜙𝑉 = 1 on 𝑉 , let 𝑓 (𝑥) :=
1𝑈 (𝑥)E𝑥 (1 − 𝜙𝑉 )∣𝑢∣ (𝑋𝜏𝑈 ) , which is bounded by condition (2.6). Note that
1 − 𝜙𝑉 = 0 on 𝑉 . Using the Lévy system of 𝑋, we have
)
[∫
(∫
]
𝜏𝑈
𝑓 (𝑥) = E𝑥
(1 − 𝜙𝑉 (𝑋𝑠 ))∣𝑢∣(𝑋𝑠 )𝑁 (𝑋𝑠 , 𝑑𝑦) 𝑑𝐻𝑠
for 𝑥 ∈ 𝐸.
0
𝐸∖𝑉
Note that the Revuz measure for PCAF
)
∫ 𝑡∧𝜏𝑈 (∫
(1 − 𝜙𝑉 (𝑋𝑠 ))∣𝑢∣(𝑦)𝑁 (𝑋𝑠 , 𝑑𝑦) 𝑑𝐻𝑠
𝑡 →
of 𝑋 𝑈 is 𝜇 :=
condition (2.4)
(∫
0
𝐸∖𝑉
)
(1
−
𝜙
(𝑥))∣𝑢∣(𝑥)𝑁
(𝑥,
𝑑𝑦)
𝑑𝜇𝐻 , and so 𝑓 = 𝐺𝑈 𝜇. Since by
𝑉
𝐸∖𝑉
)
∫ (∫
𝜇(𝑈 ) =
𝑈
𝐸∖𝑉
𝑈
𝐸∖𝑉
∫ (∫
≤
(1 − 𝜙𝑉 (𝑦))∣𝑢(𝑦)∣𝑁 (𝑥, 𝑑𝑦) 𝜇𝐻 (𝑑𝑥)
)
∣𝑢(𝑦)∣𝑁 (𝑥, 𝑑𝑦) 𝜇𝐻 (𝑑𝑥) < ∞,
∫
we have 𝑈 𝐺𝑈 𝜇(𝑥)𝜇(𝑑𝑥) ≤ ∥𝑓 ∥∞ 𝜇(𝑈 ) < ∞. Applying Lemma 2.4 to 𝑋 𝑈 yields
□
that 𝑓 ∈ ℱ𝑒𝑈 .
Lemma 2.5. Let 𝐷 be an open subset of 𝐸. Every 𝑢 ∈ ℱ𝑒 that is locally bounded
on 𝐷 satisfies conditions (2.4) and (2.5).
Proof. Let 𝑢 ∈ ℱ𝑒 be locally bounded on 𝐷. For any relatively compact open sets
𝑈, 𝑉 with 𝑈 ⊂ 𝑉 ⊂ 𝑉 ⊂ 𝐷, take 𝜙 ∈ ℱ ∩ 𝐶𝑐 (𝐷) such that 𝜙 = 1 on 𝑈 and 𝜙 = 0
on 𝑉 𝑐 . Then 𝑢𝜙 ∈ ℱ𝑒 and
∫
∫
2
𝑢(𝑦)2 𝐽(𝑑𝑥, 𝑑𝑦) =
((1 − 𝜙)𝑢)(𝑥) − ((1 − 𝜙)𝑢)(𝑦)) 𝐽(𝑑𝑥, 𝑑𝑦)
𝑈×(𝐸∖𝑉 )
𝑈×(𝐸∖𝑉 )
≤ 2ℰ(𝑢 − 𝑢𝜙, 𝑢 − 𝑢𝜙) < ∞.
This together with (2.3) implies that
∫
∫
(
)
1
1 + 𝑢(𝑦)2 𝐽(𝑑𝑥, 𝑑𝑦) < ∞.
∣𝑢(𝑦)∣𝐽(𝑑𝑥, 𝑑𝑦) ≤
2 𝑈×(𝐸∖𝑉 )
𝑈×(𝐸∖𝑉 )
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ON NOTIONS OF HARMONICITY
3503
Let 𝜙𝑉 ∈ ℱ ∩ 𝐶𝑐 (𝐷) be such that 0 ≤ 𝜙𝑉 ≤ 1 with 𝜙𝑉 = 1 on 𝑉 . Note that
∣𝑢∣ ∈ ℱ𝑒 is locally bounded on 𝐷, and so (1 − 𝜙𝑉 )∣𝑢∣ = ∣𝑢∣ − 𝜙𝑉 ∣𝑢∣ ∈ ℱ𝑒 . Thus it
follows from [4, Theorem 3.4.8] or [9, Theorem 4.6.5] that
[(
) )
[(
]
)
]
1𝑈 (𝑥)E𝑥 (1 − 𝜙𝑉 ∣𝑢∣ (𝑋𝜏𝑈 ) = E𝑥 (1 − 𝜙𝑉 )∣𝑢∣ (𝑋𝜏𝑈 ) − (1 − 𝜙𝑉 )∣𝑢∣ ∈ ℱ𝑒𝑈 .
□
Lemma 2.6. Let 𝐷 be a relatively compact open set of 𝐸. Suppose 𝑢 is a function
𝐷
in ℱloc
that is locally bounded on 𝐷 and satisfies condition (2.4). Then for every
𝑣 ∈ 𝐶𝑐 (𝐷) ∩ ℱ, the expression
∫
∫
1 𝑐
1
𝜇⟨𝑢,𝑣⟩ (𝐷) +
(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))𝐽(𝑑𝑥, 𝑑𝑦) +
𝑢(𝑥)𝑣(𝑥)𝜅(𝑑𝑥)
2
2 𝐸×𝐸
𝐷
is well defined and finite; it will still be denoted as ℰ(𝑢, 𝑣).
Proof. Clearly the first and third terms are well defined and finite. To see that the
second term is also well defined, let 𝑈 be a relatively compact open subset of 𝐷
𝐷
such that supp[𝑣] ⊂ 𝑈 . Since 𝑢 ∈ ℱloc
, there is 𝑓 ∈ ℱ so that 𝑢 = 𝑓 𝑚-a.e. and
hence q.e. on 𝑈 . Under condition (2.4),
∫
∣(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))∣𝐽(𝑑𝑥, 𝑑𝑦)
∫𝐸×𝐸
∫
≤
∣(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))∣𝐽(𝑑𝑥, 𝑑𝑦) + 2
∣𝑢(𝑥)𝑣(𝑥)∣𝐽(𝑑𝑥, 𝑑𝑦)
𝑈×𝑈
𝑈×(𝐸∖𝑈)
∫
∫
+2
∣𝑣(𝑥)∣
∣𝑢(𝑦)∣𝐽(𝑑𝑥, 𝑑𝑦)
𝑈
𝐸∖𝑈
∫
≤
∣(𝑓 (𝑥) − 𝑓 (𝑦))(𝑣(𝑥) − 𝑣(𝑦))∣𝐽(𝑑𝑥, 𝑑𝑦) + 2∥𝑢𝑣∥∞ 𝐽(supp[𝑣], 𝑈 𝑐 )
𝑈×𝑈
∫
+ 2∥𝑣∥∞
∣𝑢(𝑦)∣𝐽(𝑑𝑥, 𝑑𝑦)
< ∞.
supp[𝑣]×(𝐸∖𝑈)
In the last inequality we used (2.3) and the fact that 𝑓, 𝑣 ∈ ℱ. This proves the
lemma.
□
𝐷
is locally
Theorem 2.7. Let 𝐷 be an open subset of 𝐸. Suppose that 𝑢 ∈ ℱloc
bounded on 𝐷 satisfying conditions (2.4)-(2.5) and that
(2.10)
ℰ(𝑢, 𝑣) = 0
for every 𝑣 ∈ 𝐶𝑐 (𝐷) ∩ ℱ.
Then 𝑢 is harmonic in 𝐷. If 𝑈 is a relatively compact open subset of 𝐷 so that
P𝑥 (𝜏𝑈 < ∞) > 0 for q.e. 𝑥 ∈ 𝑈 , then 𝑢(𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )] for q.e. 𝑥 ∈ 𝑈 .
Proof. Take 𝜙 ∈ 𝐶𝑐 (𝐷)∩ℱ such that 0 ≤ 𝜙 ≤ 1 and 𝜙 = 1 in an open neighborhood
𝑉 of 𝑈 . Then 𝜙𝑢 ∈ ℱ 𝐷 . So by [4, Theorem 3.4.8] or [9, Theorem 4.6.5], ℎ1 (𝑥) :=
E𝑥 [(𝜙𝑢)(𝑋𝜏𝑈 )] ∈ ℱ𝑒 and 𝜙𝑢 − ℎ1 ∈ ℱ𝑒𝑈 . Moreover
(2.11)
ℰ(ℎ1 , 𝑣) = 0
for every 𝑣 ∈ ℱ𝑒𝑈 .
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3504
ZHEN-QING CHEN
Let ℎ2 (𝑥) := E𝑥 [((1 − 𝜙)𝑢)(𝑋𝜏𝑈 )], which is well defined by condition (2.5). Note
that by the Lévy system of 𝑋,
[(
)
]
𝑓 (𝑥) := 1𝑈 (𝑥)E𝑥 (1 − 𝜙)∣𝑢∣ (𝑋𝜏𝑈 )
)
[∫
(∫
]
𝜏𝑈
(
)
(1 − 𝜙)∣𝑢∣ (𝑧)𝑁 (𝑋𝑠 .𝑑𝑧) 𝑑𝐻𝑠 .
= 1𝑈 (𝑥) E𝑥
Define 𝜇(𝑑𝑥) := 1𝐷 (𝑥)
0
(∫
(
𝐸∖𝑉
𝐸∖𝑉
)
)
(1 − 𝜙)∣𝑢∣ (𝑧)𝑁 (𝑋𝑠 .𝑑𝑧) 𝜇𝐻 (𝑑𝑥), which is a smooth
measure of 𝑋 𝑈 . In the following, for a smooth measure 𝜈 of 𝑋 𝑈 , we will use 𝐺𝑈 𝜈
to denote E𝑥 [𝐴𝜈𝜏𝑈 ], where 𝐴𝜈 is the PCAF of 𝑋 𝑈 with Revuz measure 𝜈. Using
such a notation, 𝑓 = 𝐺𝑈 𝜇. We claim that 1𝑈 ℎ2 ∈ ℱ𝑒𝑈 and, for 𝑣 ∈ ℱ𝑒𝑈 ,
(∫
)
∫
(
)
(2.12)
ℰ(1𝐷 ℎ2 , 𝑣) =
(1 − 𝜙)𝑢 (𝑧)𝑁 (𝑋𝑠 , 𝑑𝑧) 𝜇𝐻 (𝑑𝑥).
𝑣(𝑥)1𝑈 (𝑥)
𝐸
Define
𝐸∖𝑉
(∫
𝜇1 (𝑑𝑥) :=
1𝐷 (𝑥)
𝐸∖𝑉
(∫
𝜇2 (𝑑𝑥) :=
1𝐷 (𝑥)
𝐸∖𝑉
)
(
)
+
(1 − 𝜙)𝑢 (𝑧)𝑁 (𝑋𝑠 .𝑑𝑧) 𝜇𝐻 (𝑑𝑥),
)
(
)
(1 − 𝜙)𝑢− (𝑧)𝑁 (𝑋𝑠 .𝑑𝑧) 𝜇𝐻 (𝑑𝑥).
Observe that
[
]
𝐺𝑈 𝜇1 (𝑥) = E𝑥 ((1 − 𝜙)𝑢+ )(𝑋𝜏𝑈 )
and
[
]
𝐺𝑈 𝜇2 (𝑥) = E𝑥 ((1 − 𝜙)𝑢− )(𝑋𝜏𝑈 )
for 𝑥 ∈ 𝑈.
Clearly 𝐺𝑈 𝜇1 ≤ 𝐺𝑈 𝜇. For 𝑗 ≥ 1, let 𝐹𝑗 := {𝑥 ∈ 𝑈 : 𝐺𝑈 𝜇1 (𝑥) ≤ 𝑗}, which is a finely
closed subset of 𝑈 . Define 𝜈𝑗 := 1𝐹𝑗 𝜇1 . Then for 𝑥 ∈ 𝐹𝑗 , 𝐺𝑈 𝜈𝑗 (𝑥) ≤ 𝐺𝑈 𝜇1 (𝑥) ≤ 𝑗,
while for 𝑥 ∈ 𝑈 ∖ 𝐹𝑗 ,
[
]
𝐺𝑈 𝜈𝑗 (𝑥) = E𝑥 𝐺𝑈 𝜈𝑗 (𝑋𝜎𝐹𝑗 ) ≤ 𝑗.
In other words, we have 𝐺𝑈 𝜈𝑗 ≤ 𝑗 ∧ 𝐺𝑈 𝜇1 ≤ 𝑗 ∧ 𝑓 . As both 𝐺𝑈 𝜈𝑗 and 𝑗 ∧ 𝑓
are excessive functions of 𝑋 𝑈 and 𝑚(𝑈 ) < ∞, we have by [4, Theorem 1.1.5 and
Lemma 1.2.3] that {𝐺𝑈 𝜈𝑗 , 𝑗 ∧ 𝐺𝑈 𝜇} ⊂ ℱ 𝑈 and
ℰ(𝐺𝑈 𝜈𝑗 , 𝐺𝑈 𝜈𝑗 ) ≤ ℰ(𝑗 ∧ 𝑓, 𝑗 ∧ 𝑓 ) ≤ ℰ(𝑓, 𝑓 ) < ∞.
Moreover, for each 𝑗 ≥ 1, we have by [4, Theorem 4.1.1] or [9, Theorem 5.1.3] that
∫
1
ℰ(𝐺𝑈 𝜈𝑗 , 𝐺𝑈 𝜈𝑗 ) = lim
𝐺𝑈 (𝜈𝑗 (𝑥) − 𝑃𝑡𝑈 𝐺𝑈 𝜈𝑗 (𝑥))𝐺𝑈 𝜈𝑗 (𝑥)𝑚(𝑑𝑥)
𝑡→0 𝑡 𝐸
∫
[ 𝜈𝑗 ]
1
𝐺𝑈 𝜈𝑗 (𝑥)𝑚(𝑑𝑥)
= lim
E𝑥 𝐴𝑡∧𝜏
𝑈
𝑡→0 𝑡 𝐸
∫
𝐺𝑈 𝜈𝑗 (𝑥) 1𝐹𝑗 (𝑥)𝜇1 (𝑑𝑥),
=
𝑈
∫
∫
which increases to 𝑈 𝐺𝑈 𝜇1 (𝑥)𝜇1 (𝑑𝑥). Consequently, 𝑈 𝐺𝑈 𝜇1 (𝑥)𝜇1 (𝑑𝑥) ≤ ℰ(𝑓, 𝑓 )
< ∫∞. So we have by Lemma 2.4 applied to 𝑋 𝑈 that 𝐺𝑈 𝜇1 ∈ ℱ𝑒𝑈 with ℰ(𝐺𝑈 𝜇1 , 𝑣)
= ∫𝑈 𝑣(𝑥)𝜇1 (𝑑𝑥) for every 𝑣 ∈ ℱ𝑒𝑈 . Similarly we have 𝐺𝑈 𝜇2 ∈ ℱ𝑒𝑈 with ℰ(𝐺𝑈 𝜇2 , 𝑣)
= 𝑈 𝑣(𝑥)𝜇2 (𝑑𝑥) for every 𝑣 ∈ ℱ𝑒𝑈 . It follows that 1𝑈 ℎ2 = 𝐺𝑈 𝜇1 − 𝐺𝑈 𝜇2 ∈ ℱ𝑒𝑈 ,
and claim (2.12) is established.
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ON NOTIONS OF HARMONICITY
3505
As ℎ2 = 1𝑈 ℎ2 + (1 − 𝜙)𝑢 and (1 − 𝜙)𝑢 satisfies condition (2.4), we have by
Lemma 2.6 and (2.12) that for every 𝑣 ∈ 𝐶𝑐 (𝑈 ) ∩ ℱ,
(2.13)
ℰ(ℎ2 , 𝑣) = ℰ(1𝑈 ℎ2 , 𝑣) + ℰ((1 − 𝜙)𝑢, 𝑣)
∫
=
𝑣(𝑥)(1 − 𝜙(𝑦))𝑢(𝑦)𝑁 (𝑥, 𝑑𝑦)𝜇𝐻 (𝑑𝑥)
𝐸×𝐸
∫
𝑣(𝑥)(1 − 𝜙(𝑦))𝑢(𝑦)𝑁 (𝑥, 𝑑𝑦)𝜇𝐻 (𝑑𝑥)
−
= 0.
𝐸×𝐸
This combined with (2.11) and condition (2.10) proves that
(2.14)
ℰ(𝑢 − ℎ1 − ℎ2 , 𝑣) = 0
for every 𝑣 ∈ 𝐶𝑐 (𝑈 ) ∩ ℱ.
Since 𝑢 − (ℎ1 + ℎ2 ) = (𝜙𝑢 − ℎ1 ) − 1𝐷 ℎ2 ∈ ℱ𝑒𝑈 and 𝐶𝑐 (𝑈 ) ∩ ℱ is ℰ-dense in ℱ𝑒𝑈 , the
above display holds for every 𝑣 ∈ ℱ𝑒𝑈 . In particular, we have
(2.15)
ℰ(𝑢 − ℎ1 − ℎ2 , 𝑢 − ℎ1 − ℎ2 ) = 0.
By Lemma 2.2, 𝑢(𝑋𝑡 )−ℎ1 (𝑋𝑡 )−ℎ2 (𝑋𝑡 ) is a bounded P𝑥 -martingale for q.e. 𝑥 ∈ 𝐸.
As
ℎ1 (𝑥) + ℎ2 (𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )]
for 𝑥 ∈ 𝑈,
the above implies that 𝑡 → 𝑢(𝑋𝑡∧𝜏𝑈 ) is a uniformly integrable P𝑥 -martingale for
q.e. 𝑥 ∈ 𝑈 . If P𝑥 (𝜏𝑈 < ∞) > 0 for q.e. 𝑥 ∈ 𝑈 , applying Lemma 2.2 to the Dirichlet
form (ℰ, ℱ 𝑈 ), we have 𝑢 − ℎ1 − ℎ2 = 0 q.e. on 𝑈 , and so 𝑢(𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )] for
q.e. 𝑥 ∈ 𝑈 . This completes the proof of the theorem.
□
Remark 2.8.
(i) The principal difficulty in the above proof is establishing
𝐷
satisfying con(2.14) and that 𝑢 − (ℎ1 + ℎ2 ) ∈ ℱ𝑒𝑈 for general 𝑢 ∈ ℱloc
ditions (2.4) and (2.5). If 𝑢 is assumed a priori to be in ℱ𝑒 , these facts
and therefore the theorem itself are then much easier to establish. Note
that when 𝑢 ∈ ℱ𝑒 , it follows immediately from [4, Theorem 3.4.8] or [9,
Theorem 4.6.5] that ℎ1 + ℎ2 = E𝑥 [𝑢(𝑋𝜏𝑈 )] ∈ ℱ𝑒 enjoys property (2.14)
and 𝑢 − (ℎ1 + ℎ2 ) ∈ ℱ𝑒𝑈 . Therefore (2.15) holds, and consequently 𝑢 is
harmonic in 𝐷.
(ii) If we assume that the process 𝑋 (or equivalently (ℰ, ℱ)) is 𝑚-irreducible
and that 𝑈 𝑐 is not 𝑚-polar, then P𝑥 (𝜏𝑈 < ∞) > 0 for q.e. 𝑥 ∈ 𝑈 (cf. [4,
Theorem 3.5.6] or [9]).
Theorem 2.9. Suppose 𝐷 is an open set of 𝐸 with 𝑚(𝐷) < ∞ and 𝑢 is a function
on 𝐸 satisfying condition (2.4) so that 𝑢 ∈ 𝐿∞ (𝐷; 𝑚) and {𝑢(𝑋𝑡∧𝜏𝐷 ), 𝑡 ≥ 0} is a
uniformly integrable P𝑥 -martingale for q.e. 𝑥 ∈ 𝐸. Then
(2.16)
𝐷
𝑢 ∈ ℱloc
and
ℰ(𝑢, 𝑣) = 0
for every 𝑣 ∈ 𝐶𝑐 (𝐷) ∩ ℱ.
Proof. As for q.e. 𝑥 ∈ 𝐸, {𝑢(𝑋𝑡∧𝜏𝐷 ), 𝑡 ≥ 0} is a uniformly integrable P𝑥 -martingale,
𝑢(𝑋𝑡∧𝜏𝐷 ) converges in 𝐿1 (P𝑥 ) as well as P𝑥 -a.s. to some random variable 𝜉 as
𝑡 → ∞. By considering 𝜉 + , 𝜉 − and 𝑢+ := E𝑥 [𝜉 + ], 𝑢− (𝑥) := E𝑥 [𝜉 − ] separately, we
may and do assume without loss of generality that 𝑢 ≥ 0. Note that 𝜉1{𝜏𝐷 <∞} =
𝑢(𝑋𝜏𝐷 ). Define 𝑢1 (𝑥) := E𝑥 [𝑢(𝑋𝜏𝐷 )] and 𝑢2 (𝑥) := E𝑥 [𝜉1{𝜏𝐷 =∞} ] = 𝑢 − 𝑢1 .
Let {𝑃𝑡𝐷 , 𝑡 ≥ 0} denote the transition semigroup of the subprocess 𝑋 𝐷 . Then
for q.e. 𝑥 ∈ 𝐷 and every 𝑡 > 0, by the Markov property of 𝑋 𝐷 ,
[
]
𝑃𝑡𝐷 𝑢2 (𝑥) = E𝑥 [𝑢2 (𝑋𝑡 ), 𝑡 < 𝜏𝐷 ] = E𝑥 𝜉1{𝜏𝐷 =∞} ⋅ 𝜃𝑡 , 𝑡 < 𝜏𝐷 = 𝑢2 (𝑥).
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3506
ZHEN-QING CHEN
Since 𝑢2 ∈ 𝐿2 (𝐷; 𝑚), by (2.1)-(2.2)
𝑢2 ∈ ℱ 𝐷
(2.17)
with ℰ(𝑢2 , 𝑢2 ) = 0.
On the other hand,
𝑃𝑡𝐷 𝑢(𝑥) = E𝑥 [𝑢(𝑋𝑡 ), 𝑡 < 𝜏𝐷 ] = E𝑥 [𝑢(𝑋𝜏𝐷 ), 𝑡 < 𝜏𝐷 ] ≤ 𝑢(𝑥).
Let {𝐷
∪𝑛 , 𝑛 ≥ 1} be an increasing sequence of relatively compact open subsets of 𝐷
with 𝑛≥1 𝐷𝑛 = 𝐷 and define
}
{
𝜎𝑛 := inf 𝑡 ≥ 0 : 𝑋𝑡𝐷 ∈ 𝐷𝑛 .
Let 𝑒𝑛 (𝑥) = E𝑥 [𝑒−𝜎𝑛 ], 𝑥 ∈ 𝐷, be the 1-equilibrium potential of 𝐷𝑛 with respect
to the subprocess 𝑋 𝐷 . Clearly 𝑒𝑛 ∈ ℱ 𝐷 is 1-excessive with respect to the process
𝑋 𝐷 , 𝑒𝑛 (𝑥) = 1 q.e. on 𝐷𝑛 . Let 𝑎 := ∥1𝐷 𝑢∥∞ . Then for every 𝑡 > 0,
𝑒−𝑡 𝑃𝑡𝐷 ((𝑎𝑒𝑛 ) ∧ 𝑢)(𝑥) ≤ ((𝑎𝑒𝑛 ) ∧ 𝑢)(𝑥)
for q.e. 𝑥 ∈ 𝐷.
By [4, Lemma 1.2.3] or [10, Lemma 8.7], we have (𝑎𝑒𝑛 ) ∧ 𝑢 ∈ ℱ 𝐷 for every 𝑛 ≥ 1.
𝐷
Since (𝑎𝑒𝑛 ) ∧ 𝑢 = 𝑢 𝑚-a.e. on 𝐷𝑛 , we have 𝑢 ∈ ℱloc
.
Let 𝑈 be a relatively compact open subset of 𝐷. Let 𝜙 ∈ 𝐶𝑐 (𝐷) ∩ ℱ so that
0 ≤ 𝜙 ≤ 1 and 𝜙 = 1 in an open neighborhood 𝑉 of 𝑈 . Define for 𝑥 ∈ 𝐸,
ℎ1 (𝑥) := E𝑥 [(𝜙𝑢)(𝑋𝜏𝑈 )]
and ℎ2 (𝑥) := E𝑥 [((1 − 𝜙)𝑢)(𝑋𝜏𝑈 )] .
Then 𝑢1 = ℎ1 + ℎ2 on 𝐸. Since 𝜙𝑢 ∈ ℱ, we know as in (2.11) that ℎ1 ∈ ℱ𝑒 and
ℰ(ℎ1 , 𝑣) = 0
for every 𝑣 ∈ ℱ𝑒𝑈 .
By the same argument as that for (2.13), we have
ℰ(ℎ2 , 𝑣) = 0
for every 𝑣 ∈ ℱ𝑒𝑈 .
These together with (2.17) in particular imply that
ℰ(𝑢, 𝑣) = ℰ(ℎ1 + ℎ2 + 𝑢2 , 𝑣) = 0
for every 𝑣 ∈ 𝐶𝑐 (𝑈 ) ∩ ℱ.
Since 𝑈 is an arbitrary relatively compact subset of 𝐷, we have
ℰ(𝑢, 𝑣) = 0
for every 𝑣 ∈ 𝐶𝑐 (𝐷) ∩ ℱ.
This completes the proof.
□
Remark 2.10. As mentioned in the Introduction, the principal difficulty for the
proof of the above theorem is establishing that a function 𝑢 harmonic in 𝐷 is in
𝐷
with ℰ(𝑢, 𝑣) = 0 for every 𝑣 ∈ ℱ ∩ 𝐶𝑐 (𝐷). If a priori 𝑢 is assumed to be in ℱ𝑒 ,
ℱloc
then Theorem 2.9 is easy to establish. In this case, it follows from [4, Theorem 3.4.8]
or [9, Theorem 4.6.5] that 𝑢1 = ℎ1 + ℎ2 = E𝑥 [𝑢(𝑋𝜏𝑈 )] ∈ ℱ𝑒 and that the second
property of (2.14) holds. This together with (2.17) immediately implies that 𝑢
enjoys (2.16). (See also Proposition 2.5 of [1] for this simple case but under an
additional assumption that 1 ∈ ℱ with ℰ(1, 1) = 0.)
Combining Theorems 2.7 and 2.9, we have the following.
Theorem 2.11. Let 𝐷 be an open subset of 𝐸. Suppose that 𝑢 is a function on 𝐸
that is locally bounded on 𝐷 and satisfies conditions (2.4) and (2.5). Then
(i) 𝑢 is harmonic in 𝐷 if and only if condition (2.16) holds.
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ON NOTIONS OF HARMONICITY
3507
(ii) Assume that for every relatively compact open subset 𝑈 of 𝐷, P𝑥 (𝜏𝑈 < ∞)
> 0 for q.e. 𝑥 ∈ 𝑈 . (By Remark 2.8(ii), this condition is satisfied if (ℰ, ℱ)
is 𝑚-irreducible.) Then 𝑢 is harmonic in 𝐷 if and only if for every relatively compact subset 𝑈 of 𝐷, 𝑢(𝑋𝜏𝑈 ) ∈ 𝐿1 (P𝑥 ) and 𝑢(𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )]
for q.e. 𝑥 ∈ 𝑈 .
Example 2.12 (Stable-like process on ℝ𝑑 ). Consider the following Dirichlet form
(ℰ, ℱ) on 𝐿2 (ℝ𝑑 , 𝑑𝑥), where
{
𝛼/2,2
𝑑
(ℝ ) := 𝑢 ∈ 𝐿2 (ℝ𝑑 ; 𝑑𝑥) :
ℱ = 𝑊
}
∫
1
2
(𝑢(𝑥) − 𝑢(𝑦))
𝑑𝑥𝑑𝑦 < ∞ ,
∣𝑥 − 𝑦∣𝑑+𝛼
ℝ𝑑 ×ℝ𝑑
∫
1
𝑐(𝑥, 𝑦)
ℰ(𝑢, 𝑣) =
(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))
𝑑𝑥𝑑𝑦
for 𝑢, 𝑣 ∈ ℱ.
2 ℝ𝑑 ×ℝ𝑑
∣𝑥 − 𝑦∣𝑑+𝛼
Here 𝑑 ≥ 1, 𝛼 ∈ (0, 2) and 𝑐(𝑥, 𝑦) is a symmetric function in (𝑥, 𝑦) that is bounded
between two positive constants. In literature, 𝑊 𝛼,2 (ℝ𝑑 ) is called the Sobolev space
on ℝ𝑑 of fractional order (𝛼/2, 2). For an open set 𝐷 ⊂ ℝ𝑑 , 𝑊 𝛼,2 (𝐷) is similarly
defined as above but with 𝐷 in place of ℝ𝑑 . It is easy to check that (ℰ, ℱ) is a regular
Dirichlet form on 𝐿2 (ℝ𝑑 ; 𝑑𝑥), and its associated symmetric Hunt process 𝑋 is called
a symmetric 𝛼-stable-like process on ℝ𝑑 , which is studied in [5]. The process 𝑋 has
strictly positive jointly continuous transition density function 𝑝(𝑡, 𝑥, 𝑦) and hence
is irreducible. Moreover, there is a constant 𝑐 > 0 such that
(2.18)
𝑝(𝑡, 𝑥, 𝑦) ≤ 𝑐 𝑡−𝑑/𝛼
for 𝑡 > 0 and 𝑥, 𝑦 ∈ ℝ𝑑 ,
and consequently by [8, Theorem 1],
(2.19)
sup E𝑥 [𝜏𝑈 ] < ∞
𝑥∈𝑈
for any open set 𝑈 having finite Lebesgue measure. When 𝑐(𝑥, 𝑦) is constant, the
process 𝑋 is nothing but the rotationally symmetric 𝛼-stable process on ℝ𝑑 . In
this example, the jumping measure
𝑐(𝑥, 𝑦)
𝐽(𝑑𝑥, 𝑑𝑦) =
𝑑𝑥𝑑𝑦.
∣𝑥 − 𝑦∣𝑑+𝛼
Hence for any non-empty open set 𝐷 ⊂ ℝ𝑑 , condition (2.4) is satisfied if and only if
(1∧∣𝑥∣−𝑑−𝛼 )𝑢(𝑥) ∈ 𝐿1 (ℝ𝑑 ). Moreover, for such a function 𝑢 and relatively compact
open sets 𝑈, 𝑉 with 𝑈 ⊂ 𝑉 ⊂ 𝑉 ⊂ 𝐷, by the Lévy system of 𝑋,
) ]
[∫ 𝜏𝑈 (∫
𝑐(𝑋𝑠 , 𝑦) ∣𝑢(𝑋𝑠 )∣
𝑑𝑦 𝑑𝑠
sup E𝑥 [(1𝑉 𝑐 ∣𝑢∣)(𝑋𝜏𝑈 )] = sup E𝑥
∣𝑋𝑠 − 𝑦∣𝑑+𝛼
𝑥∈𝑈
𝑥∈𝑈
0
𝑉𝑐
( ∫
)
≤
𝑐
(2.20)
(1 ∧ ∣𝑦∣−𝑑−𝛼 )∣𝑢(𝑦)∣𝑑𝑦 sup E𝑥 [𝜏𝑈 ] < ∞.
ℝ𝑑
𝑥∈𝑈
In other words, for this example, condition (2.6) and hence (2.5) is a consequence
of (2.4). So Theorem 2.9 says that for an open set 𝐷 and a function 𝑢 on ℝ𝑑
that is locally bounded on 𝐷 with (1 ∧ ∣𝑥∣−𝑑−𝛼 )𝑢(𝑥) ∈ 𝐿1 (ℝ𝑑 ), the following are
equivalent:
(i) 𝑢 is harmonic in 𝐷;
(ii) for every relatively compact subset 𝑈 of 𝐷, 𝑢(𝑋𝜏𝑈 ) ∈ 𝐿1 (P𝑥 ) and 𝑢(𝑥) =
E𝑥 [𝑢(𝑋𝜏𝑈 )] for q.e. 𝑥 ∈ 𝑈 ;
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3508
ZHEN-QING CHEN
∫
𝛼,2
𝐷
(iii) 𝑢 ∈ ℱloc
= 𝑊loc
(𝐷) and
ℝ𝑑 ×ℝ𝑑
(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))
𝑐(𝑥, 𝑦)
𝑑𝑥𝑑𝑦 = 0
∣𝑥 − 𝑦∣𝑑+𝛼
for every 𝑣 ∈ 𝐶𝑐 (𝐷) ∩ 𝑊 𝛼/2,2 (ℝ𝑑 ).
□
Example 2.13 (Diffusion process on a locally compact separable metric space).
Let (ℰ, ℱ) be a local regular Dirichlet form on 𝐿2 (𝐸; 𝑚), where 𝐸 is a locally
compact separable metric space and 𝑋 is its associated Hunt process. In this case,
𝑋 has continuous sample paths, and so the jumping measure 𝐽 is null (cf. [9]).
Hence conditions (2.4) and (2.5) are automatically satisfied. Let 𝐷 be an open
subset of 𝐸 and 𝑢 be a function on 𝐸 that is locally bounded in 𝐷. Then by
Theorem 2.11, 𝑢 is harmonic in 𝐷 if and only if condition (2.16) holds.
Now consider the following special case: 𝐸{ = ℝ𝑑 with 𝑑 ≥ 1, 𝑚(𝑑𝑥) is the
}
Lebesgue measure 𝑑𝑥 on ℝ𝑑 , ℱ = 𝑊 1,2 (ℝ𝑑 ) := 𝑢 ∈ 𝐿2 (ℝ𝑑 ; 𝑑𝑥) ∣ ∇𝑢 ∈ 𝐿2 (ℝ𝑑 ; 𝑑𝑥)
and
𝑑 ∫
∂𝑢(𝑥) ∂𝑣(𝑥)
1 ∑
𝑎𝑖𝑗 (𝑥)
𝑑𝑥
for 𝑢, 𝑣 ∈ 𝑊 1,2 (ℝ𝑑 ),
ℰ(𝑢, 𝑣) =
2 𝑖,𝑗=1 ℝ𝑑
∂𝑥𝑖 ∂𝑥𝑗
where (𝑎𝑖𝑗 (𝑥))1≤𝑖,𝑗≤𝑑 is a 𝑑 × 𝑑-matrix valued measurable function on ℝ𝑑 that is
uniformly elliptic and bounded. In literature, 𝑊 1,2 (ℝ𝑑 ) is the Sobolev space on ℝ𝑑
of order (1, 2). For an open set 𝐷 ⊂ ℝ𝑑 , 𝑊 1,2 (𝐷) is similarly defined as above but
with 𝐷 in place of ℝ𝑑 . Then (ℰ, ℱ) is a regular local Dirichlet form on 𝐿2 (ℝ𝑑 ; 𝑑𝑥),
and its associated Hunt process 𝑋 is a conservative diffusion on ℝ𝑑 having jointly
continuous transition density function. Let 𝐷 be an open set in ℝ𝑑 . Then by
Theorem 2.11, the following are equivalent for a locally bounded function 𝑢 on 𝐷:
(i) 𝑢 is harmonic in 𝐷;
(ii) for every relatively compact open subset 𝑈 of 𝐷, 𝑢(𝑋𝜏𝑈 ) ∈ 𝐿1 (P𝑥 ) and
𝑢(𝑥) = E𝑥 [𝑢(𝑋𝜏𝑈 )] for q.e. 𝑥 ∈ 𝑈 ;
𝑑 ∫
∑
∂𝑢(𝑥) ∂𝑣(𝑥)
1,2
(𝐷) and
𝑎𝑖𝑗 (𝑥)
𝑑𝑥 = 0 for every 𝑣 ∈ 𝐶𝑐 (𝐷)∩
(iii) 𝑢 ∈ 𝑊loc
∂𝑥𝑖 ∂𝑥𝑗
𝑑
𝑖,𝑗=1 ℝ
𝑊 1,2 (ℝ𝑑 ).
In fact, in this case, it can be shown that every (locally bounded) harmonic function
has a continuous version.
□
Example 2.14 (Diffusions with jumps on ℝ𝑑 ). Consider the following Dirichlet
form (ℰ, ℱ), where ℱ = 𝑊 1,2 (ℝ𝑑 ) and
𝑑 ∫
∂𝑢(𝑥) ∂𝑣(𝑥)
1 ∑
𝑎𝑖𝑗 (𝑥)
𝑑𝑥
ℰ(𝑢, 𝑣) =
2 𝑖,𝑗=1 ℝ𝑑
∂𝑥𝑖 ∂𝑥𝑗
∫
1
𝑐(𝑥, 𝑦)
+
(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))
𝑑𝑥𝑑𝑦
2 ℝ𝑑 ×ℝ𝑑
∣𝑥 − 𝑦∣𝑑+𝛼
for 𝑢, 𝑣 ∈ 𝑊 1,2 (ℝ𝑑 ). Here 𝑑 ≥ 1, (𝑎𝑖𝑗 (𝑥))1≤𝑖,𝑗≤𝑑 is a 𝑑×𝑑-matrix valued measurable
function on ℝ𝑑 that is uniformly elliptic and bounded, 𝛼 ∈ (0, 2) and 𝑐(𝑥, 𝑦) is a
symmetric function in (𝑥, 𝑦) that is bounded between two positive constants. It is
easy to check that (ℰ, ℱ) is a regular Dirichlet form on 𝐿2 (ℝ𝑑 ; 𝑑𝑥). Its associated
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ON NOTIONS OF HARMONICITY
3509
symmetric Hunt process 𝑋 has both the diffusion and jumping components. Such
a process has recently been studied in [6]. It is shown there that the process 𝑋 has
a strictly positive jointly continuous transition density function 𝑝(𝑡, 𝑥, 𝑦) and hence
is irreducible. Moreover, a sharp two-sided estimate is obtained in [6] for 𝑝(𝑡, 𝑥, 𝑦).
In particular, there is a constant 𝑐 > 0 such that
)
(
𝑝(𝑡, 𝑥, 𝑦) ≤ 𝑐 𝑡−𝑑/𝛼 ∧ 𝑡−𝑑/2
for 𝑡 > 0 and 𝑥, 𝑦 ∈ ℝ𝑑 .
Note that when (𝑎𝑖𝑗 )1≤𝑖,𝑗≤𝑑 is the identity matrix and 𝑐(𝑥, 𝑦) is constant, the
process 𝑋 is nothing but the symmetric Lévy process that is the independent sum
of a Brownian motion and a rotationally symmetric 𝛼-stable process on ℝ𝑑 . In this
example, the jumping measure
𝐽(𝑑𝑥, 𝑑𝑦) =
𝑐(𝑥, 𝑦)
𝑑𝑥𝑑𝑦.
∣𝑥 − 𝑦∣𝑑+𝛼
Hence for any non-empty open set 𝐷 ⊂ ℝ𝑑 , condition (2.4) is satisfied if and only
if (1 ∧ ∣𝑥∣−𝑑−𝛼 )𝑢(𝑥) ∈ 𝐿1 (ℝ𝑑 ). By the same reasoning as that for (2.20), we see
that for this example, condition (2.6) and hence (2.5) is implied by condition (2.4).
So Theorem 2.9 says that for an open set 𝐷 and a function 𝑢 on ℝ𝑑 that is locally
bounded on 𝐷 with (1 ∧ ∣𝑥∣−𝑑−𝛼 )𝑢(𝑥) ∈ 𝐿1 (ℝ𝑑 ), the following are equivalent:
(i) 𝑢 is harmonic in 𝐷 with respect to 𝑋;
(ii) for every relatively compact subset 𝑈 of 𝐷, 𝑢(𝑋𝜏𝑈 ) ∈ 𝐿1 (P𝑥 ) and 𝑢(𝑥) =
E𝑥 [𝑢(𝑋𝜏𝑈 )] for q.e. 𝑥 ∈ 𝑈 ;
1,2
(𝐷) such that for every 𝑣 ∈ 𝐶𝑐 (𝐷) ∩ 𝑊 1,2 (ℝ𝑑 ),
(iii) 𝑢 ∈ 𝑊loc
𝑑 ∫
∑
𝑖,𝑗=1
ℝ𝑑
𝑎𝑖𝑗 (𝑥)
∫
+
ℝ𝑑 ×ℝ𝑑
∂𝑢(𝑥) ∂𝑣(𝑥)
𝑑𝑥
∂𝑥𝑖 ∂𝑥𝑗
(𝑢(𝑥) − 𝑢(𝑦))(𝑣(𝑥) − 𝑣(𝑦))
𝑐(𝑥, 𝑦)
𝑑𝑥𝑑𝑦 = 0.
∣𝑥 − 𝑦∣𝑑+𝛼
□
Remark 2.15. It is possible to extend the results of this paper to a general 𝑚symmetric right process 𝑋 on a Lusin space, where 𝑚 is a positive 𝜎-finite measure
with full topological support on 𝐸. In this case, the Dirichlet (ℰ, ℱ) of 𝑋 is a
quasi-regular Dirichlet form on 𝐿2 (𝐸; 𝑚). By [7], (ℰ, ℱ) is quasi-homeomorphic
to a regular Dirichlet form on a locally compact separable metric space. So the
results of this paper can be extended to the quasi-regular Dirichlet form setting
by using this quasi-homeomorphism. However since the notion of an open set is
not invariant under a quasi-homeomorphism, some modifications are needed. We
need to replace the open set 𝐷 in Definition 2.1 by a quasi-open set 𝐷. Similar
modifications are needed for conditions (2.4) and (2.5) as well. We say a function
𝑢 is harmonic in a quasi-open set 𝐷 ⊂ 𝐸 if for every quasi-open subset 𝑈 ⊂ 𝐷 with
𝑈 ∩ 𝐹𝑘 ⊂ 𝐷 for every 𝑘 ≥ 1, where {𝐹𝑘 , 𝑘 ≥ 1} is an ℰ-nest consisting of compact
sets, 𝑡 → 𝑢(𝑋𝑡∧𝜏𝑈 ∩𝐹𝑘 ) is a uniformly integrable P𝑥 -martingale for q.e. 𝑥 ∈ 𝑈 ∩ 𝐹𝑘
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3510
ZHEN-QING CHEN
𝐷
and for every 𝑘 ≥ 1. The local Dirichlet space ℱloc
needs to be replaced by
{
∘
𝐷
ℱloc
=
𝑢 : there is an increasing sequence of quasi–open sets {𝐷𝑛 }
with
∞
∪
𝐷𝑛 = 𝐷 q.e. and a sequence {𝑢𝑛 } ⊂ ℱ 𝐷
𝑛=1
}
such that 𝑢 = 𝑢𝑛 𝑚-a.e. on 𝐷𝑛 .
Condition (2.16) should be replaced by
(2.21)
∘
𝐷
𝑢 ∈ℱloc
and
ℰ(𝑢, 𝑣) = 0 for every 𝑣 ∈ ℱ with ℰ-supp[𝑣] ⊂ 𝐷.
Here ℰ-supp[𝑢] is the smallest quasi-closed set that 𝑢 vanishes 𝑚-a.e. on its complement. We leave the details to interested readers.
□
Acknowledgements
The author thanks Rich Bass and Takashi Kumagai for helpful discussions. He
also thanks Rongchan Zhu for helpful comments.
References
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Department of Mathematics, University of Washington, Seattle, Washington 98195
E-mail address: [email protected]
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